On a new normalization for tractor covariant derivatives. (English) Zbl 1264.58029

Hammerl, Matthias; Somberg, Petr; Souček, Vladimír; Šilhan, Josef

J. Eur. Math. Soc. (JEMS) 14, No. 6, 1859-1883 (2012).

A parabolic geometry is a quotient \(G/P\) of a real semisimple Lie group by the action of a parabolic subgroup \(P\). More generally, a parabolic geometry of type \((G,P)\) on a manifold \(M\) is a pair consisting of a principal \(P\)-bundle \(\mathcal{G} \to M\) and a Cartan connection \(\omega \in \Omega^1 ( \mathcal{G} , \mathfrak{g})\). Examples of parabolic geometries on a manifold include CR, conformal or projective structures.
If \((\mathcal{G} , \omega)\) is a (regular and normal) parabolic geometry on a manifold \(M\), and \(\mathbb{V}\) is a \(G\)-module, the vector bundle \(V \to M\) associated to \(\mathcal{G}\) and the representation \(\mathbb V\) is called the tractor bundle. The normal Cartan connection \(\omega\) on \(\mathcal{G}\) induces a covariant derivative \(\nabla^\omega\) on \(V\).
As it is carefully explained in the detailed introduction of the paper, there exists a curved version of the Berstein-Gelfand-Gelfand (BGG) resolution of an irreducible \(G\)-module \(\mathbb V\). This resolution is made up of invariant differential operators \(D_i\). The first of these operators, \(D_0\), is overdetermined and, in the flat case \(M = G/P\), the tractor covariant derivative \(\nabla^\omega\) is already known to yield the prolongation of \(D_0\).
The main result of this paper consists of the curved analogue for such a prolongation. To prove this result, the authors develop a new normalization of \(\nabla^\omega\), which is invariantly defined and generalizes some other constructions existing in the literature.
In Section 4, the authors extend this construction to the other operators \(D_i\) in the BGG sequence. These normalizations require more complicated modifications (with differential terms) of the exterior covariant derivative \(d^{\nabla^\omega}\). As the authors mention, their approach “is based on standard BGG techniques”.
Finally, the paper includes some examples, taken from projective, conformal and Grassmannian geometry, that illustrate the results commented above.
This paper is very clearly written, in an elegant and coordinate independent manner.

Reviewer: José Navarro Garmendia (Badajoz)

MSC:

58J70	Invariance and symmetry properties for PDEs on manifolds
53A30	Conformal differential geometry (MSC2010)
53A20	Projective differential geometry
58A32	Natural bundles
53A55	Differential invariants (local theory), geometric objects

Keywords:

parabolic geometry; prolongation of overdetermined PDE’s; BGG sequence; tractor covariant derivatives

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